Cube Map Pre-Integration & Sampling

Since the papers "Real-time Shading with Filtered Importance Sampling" 1 and "Real Shading in Unreal Engine 4" 2, most people have been using the mips of cube maps to pre-integrate the specular illuminance to feed their BRDF for different values of roughness.

You often read something like "a different mip corresponds to some roughness value" but it's generally not very clear how exactly do you tie the mip level and the roughness value of the BRDF?

There are various possible schemes, some of them are explored by S. Lagarde in "Moving Frostbite to PBR" 3 in section 4.9.2 eq. 63 where they chose the simple expression:

m = \alpha \cdot N

Where:

m is the mip level

N is the maximum mip level (e.g for a 256x256 cube map, N would be log_2(256)=8)

Ignoring the Fresnel coefficient F that is factored out of the pre-integration anyway, and the shadowing/masking terms that are either view-dependent (not in the integral) or light-dependent (not relevant to the problem at hand),
we can focus on the normal distribution function D( \cos(\theta), \alpha ) for which we know that, by definition: